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PARADISE LOST?
ayeing the gate apprehensively under the stern regard of the
uncompromising Dutchman. The postulational method for
securing freedom from contradiction proposed by Hilbert will,
says, Brouwer, accomplish its end - produce no contradictions,
but 'nothing of mathematical value will be attained hi this
manner; a false theory which is not stopped by a contradiction
is none the less false, just as a criminal policy unchecked by a
reprimanding court is none the less criminal.'
The root of Brouwer's objection to the 'criminal policy' of his
opponents is something new - at least in mathematics. He
objects to an unrestricted use of Aristotelian logic, particularly
in dealing with infinite sets, and he maintains that such logic
is bound to produce contradictions when applied to sets which
cannot be definitely constructed in Kronecker's sense (a rule of
procedure must be given whereby the things in the set can be
produced). The law of 'excluded middle' (a thing must have a
certain property or must not have that property, as for example
in the assertion that a number is prime or is not prime) is
legitimately usable only when applied to finite sets. Aristotle
devised his logic as a body of working rules for finite sets,
basing his method on human experience of finite sets, and there
is no reason whatever for supposing that a logic which is ade-
quate for the finite will continue to produce consistent (not
contradictory) results when applied to the infinite. This seems
reasonable enough when we recall that the very definition of an
infinite set emphasizes that a part of an infinite set may contain
precisely as many things as the whole set (as we have illustrated
many times), a situation which never happens for a finite set
when 'part* means some, but not all (as it does in the definition
of an infinite set).
Here we have what some consider the root of the trouble in
Cantor's theory of the actual infinite. For the definition of a set
(as stated some time back), by which all things having a certain
quality are 'united' to form a 'set' (or 'class'), is not suitable as
a basis for the theory of sets, in that the definition either is not
constructive (in Kronecker's sense) or assumes a constructibility
which no mortal can produce. Brouwer claims that the use of
the law of excluded middle in such a situation is at best merely
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